"Young man, in this house we obay the laws of physics!" Homer Simpson
It does'nt mater what kind of contraption you use to get something up to speed.
Think of this problem. Put a ball trap in front of your device, run the balls back into the gun.
Do you have a reactionless drive? (i.e does the regular recoil of the balls hitting the trap now pull the whole thing forward?)
If you do you are rich beyond averace and have brought the stars into reach. I don't think you do.
I don't see the parallel you're trying to draw. How does a sling have any recoil? All that happens when you release it is the centripetal force goes to zero, which unbalances the centrifugal force and allows the projectile to fly away tangentially. There is no recoil because there is no equal and opposite force imparted instantaneously to the projectile. It's energy is kinetic at that point, having been imparted (as a moment of inertia) by the act of twirling it. I envision the same physics applying here.
But, as I've said, I might be misunderstanding the construction of this device.